1
1Short-Run CostFunction
14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen
October 15, 2007
Lecture
13
Cost
Functions
Outline
1. Chap 7: Short-Run Cost Function
2. Chap 7: Long-Run Cost Function
Cost
Function
Let wbe the cost per unit of labor and rbe the cost per unit of capital. With
the input Labor (L) and Capital (K), the production cost is w×L+ r×K.
A cost function C(q) is a function of q, which tells us what the minimum cost
is for producing qunits of output. We can also split total cost into fixed cost
and variable cost as follows:
C(q) = F C+ V C(q).
Fixed cost is independent of quantity, while variable cost is dependent on quan tity.
1
Short-Run
Cost
Function
In the short-run, firms cannot change capital, that is to say,
r×K= const.
Recall the production function given fixed capital level Kin the short run (refer
to Lecture 11) (see Figure 1). Suppose w= 1, the variable cost curve can be
derived from Figure 1. Adding r×K to the variable cost, we obtain the total
cost curve (see Figure 2). Average total cost is
T C F C+ V C rK wL(q;K)
AT C= = = + .
q q q q
With the definition of the average product of labor: q
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1Short-Run CostFunction
40 35 30 25 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 L Q
Figure 1: Short Run Production Function.
0 5 10 15 20 25 30 35 40 45 C VC TC 50 0 1 2 3 4 5 6 7 8 9 10 q
L 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 AP MP 3
1Short-Run CostFunction
we can rewrite AT Cas
rK w
AT C= + , q APL
in which the average variable cost is
V C wL(q;K) w
= = .
q q APL
Likewise, we rewrite the marginal cost:
dT C dV C dL(q) w w
M C= = = w = = .
dq dq dq ∂q M P
L ∂L
In Lecture 11, we discussed the relation between average product of labor and
marginal product of labor (see Figure 3). We draw the curves for AVC and
Figure 3: Average Product of Labor and Marginal Product of Labor.
M Cin the same way (see Figure 4). The relation between M C and AVC is:
If • M C< AV C, AVC decreases; if • M C> AV C, AVC increases;
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2Long-Run CostFunction
0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 C AVC ATC MC L
Figure 4: Average Cost, Average Variable Cost, and Marginal Cost. if
•
M C= AVC,
AVC is minimized.
Now consider the total cost. Note that the difference between AT C and AVC
decreases with q as the average fixed cost term dies out (see Figure 4). The
relation between M C and AT C is:
If • M C<AT C, AT C decreases; if • M C>AT C, AT C increases; if • M C= AT C, AT C is minimized.
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2Long-Run CostFunction
k 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 L
Figure 5: Isoquant Curve.
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Long-Run
Cost
Function
In the long-run, both K and Lare variable. The isoquant curve describes the
same output level with different combination of K and L (see Figure 5). The
slope of an isoquant curve is
M PL −M RT S= −
M PK
.
Similarly, the isocost curve is constructed by different (K, L) with the same cost (see Figure 6). The isocost curve equation is:
rK+ wL= const,
therefore, it is linear, with a slope − wr.
Now we want to minimize the costrK+wLsubject to an output level Q(K, L) =
q. This minimum cost can be obtained when the isocost curve is tangent to the isoquant curve (see Figure 7). Thus the slopes of these two curves are equal:
M PL w
M RT S= = . M PK r
Now consider an increase in wage (w). The slope of the isocost curve increases (see Figure 8), and the firm use more capital and less labor. The firm’s choice
of input moves from A to B in the figure.
The expansion path shows the minimum cost combinations of labor and capital at each level of output (see Figure 9).
K 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 L 6
2Long-Run CostFunction
Figure 6: Isocost Curve.
0 1 2 3 4 5 6 7 8 9 10
L
Figure 7: Minimize the Cost Subject to a Output Level.
0 1 2 3 4 5 6 7 8 9 10 K
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2Long-Run CostFunction
0 1 2 3 4 5 6 7 8 9 10 K A B 0 1 2 3 4 5 6 7 8 9 10 L
Figure 8: The Change of Cost Minimized Situation.
2 3 4 5 6 7 8 9 10 11 12 K Expansion Path 2 3 4 5 6 7 8 9 10 11 12 L
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2Long-Run CostFunction
Example (Calculating the Cost.). Given the production function
2 3
2 3
q= L K . In the short run,
3 2 q CSR(q;K) = rK+ w , K where K is fixed.
In the long run, according to the equation M PL w = , M PK r we have K w = . L r
Then the expansion path is
w K= L.
r
Substituting this result into the production function, we obtain r ( ) 3 4 1 2 L= q , w w ( ) 3 4 1 2 K= q . r Hence, the long-run cost function is:
√
3 4
